I want to show that if $g \ \sim N(0,1)$, for all $t>0$ we have $P(g\geq t)\leq e^{-t^2/2}$.
My solution: Let $\lambda>0$. $P(g\geq t)=P(e^{\lambda g}\geq e^{\lambda t})\leq \frac{E[e^{\lambda g}]}{e^{\lambda t}}=\frac{e^{\lambda^2/2}}{e^{\lambda t}}$.
Then substitute $\lambda=t$, then $P(g\geq t)\leq e^{-t^2/2}$.
Is it correct or I'm missing something?